Pseudo-Maximization via Method of Mixtures

Abstract

In this chapter we describe the method of mixtures to perform pseudo-maximization that generates self-normalized processes via (10.4). Section 11.1 describes a prototypical example and Laplace's method for asymptotic evaluation of integrals. Section 11.2 reviews the method of mixtures used by Robbins and Siegmund (1970) to evaluate boundary crossing probabilities for Brownian motion, and generalizes the method to analyze boundary crossing probabilities for self-normalized processes. In Sect. 11.3 we describe a class of mixing density functions that are particularly useful for developing L_p and exponential inequalities for self-normalized processes, details of which are given in the next chapter.